Introduction to Riemann sums

1. Indefinite
Integral
Definite
Integral
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2. Indefinite Integral versus Definite Integral
A family of functions The area under a curve
(A number)
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3. Archimedes
http://www.philosophyprofessor.com/images/philosophers/archimedes.jpg
Born: About 287 BC in Syracuse, Sicily.
Died: 212 or 211 BC in Syracuse.
Generally regarded as the greatest mathematician and scientist of antiquity.
He is called the "father of integral calculus" and also the "father of mathematical physics".
http://www.cs.drexel.edu/~crorres/Archimedes/contents.html
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4. "Give me a lever long enough
and a fulcrum strong enough,
and single­handedly I will move the world."
­ Archimedes
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5. Method of Exhaustion
http://math.furman.edu/~dcs/java/circle.html
Does this idea of inscribing and circumscribing polygons
to close in on the area of a circle remind you of a theorem
we have studied before?
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6. 6

7. Review:
Ways to approximate the definite integral:
1. Counting Squares
Images from: Math 122 ­ Calculus for Biology II, Fall Semester, 2004, Riemann Sums and Numerical Integration, SDSU & Joseph M. Mahaffy
2. Trapezoidal Rule
http://www.csun.edu/~hcmth018/Trapez.html
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8. 8

9. http://www.ecs.fullerton.edu/~mathews/a2001/Animations/Quadrature/Quadrature.html
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10. Answer Key
L6 = = 1.2178...
U6 = = 1.5928...
1 4
M6 = = 1.3769...
1 4
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11. M6 is an underestimate.
Because the function is decreasing more
rapidly on the left side of the midpoint
than on the right side of the midpoint,
there is more area left out and less area
included. (See figure.)
T6 = 1.4053...
T6 is an overestimate for the actual area
under the curve. (See figure)
All of T6 error is outside the curve.
M6 error is both inside and outside of the curve (with more area
excluded than included)
So M6 is a more precise estimate of the true area.
sample point =
ci = 1, π/2, 2, 3, 4, 6.
sample point =
ci = 0, 1, 3, 4, 3π/2, 5.
c2 c5
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